If the transmission line of Problem 7.6 is short-circuited,...

If the transmission line of Problem $7.6$ is short-circuited, show that its input impedance is given by $$ Z_{s c}=Z_{0} \tanh \gamma l $$ and when it is open-circuited the input impedance is $$ Z_{0 c}=Z_{0} \operatorname{coth} \gamma l $$ By taking the product of these quantities, suggest a method for measuring the characteristic impedance of the line.

If the transmission line of Problem $7.6$ is short-circuited, show that its input impedance is given by
$$
Z_{s c}=Z_{0} \tanh \gamma l
$$
and when it is open-circuited the input impedance is
$$
Z_{0 c}=Z_{0} \operatorname{coth} \gamma l
$$
By taking the product of these quantities, suggest a method for measuring the characteristic impedance of the line.

Key Concepts

Transmission Line Theory

This concept deals with the analysis of how electrical signals propagate along conductors which have distributed inductance, capacitance, resistance, and conductance. The behavior of signals on a transmission line, including attenuation and phase shift, is described by parameters such as the characteristic impedance and the propagation constant.

Characteristic Impedance

The characteristic impedance of a transmission line is a fundamental parameter defined as the impedance that the line would exhibit if it were infinitely long. It is critical for ensuring maximum power transfer and minimum reflections in a system. In the context of the problem, it is shown that the product of the input impedances under different load conditions equals the square of the characteristic impedance, providing a method to measure it.

Input Impedance

Input impedance refers to the impedance presented by a transmission line when observed from a given point, typically the feed point. It depends on the termination of the line (open, short, or any other load), the line’s length, and the propagation characteristics of the medium. In the problem, the input impedance is calculated for both short-circuited and open-circuited conditions.

Boundary Conditions (Short-circuit and Open-circuit)

These conditions refer to the electrical terminations at the end of a transmission line. A short circuit forces the voltage to zero at the termination while an open circuit forces the current to zero. Each condition leads to a different expression for the input impedance, as derived using transmission line equations involving hyperbolic functions.

Hyperbolic Functions in Transmission Lines

Hyperbolic functions such as tanh and coth appear in the solutions of the transmission line equations, particularly when dealing with lossy lines. They provide a convenient way to express the variation of voltage and current along the line as a function of the complex propagation constant and the line length. In this problem, tanh and coth help express the input impedances for short-circuit and open-circuit terminations respectively.

Measurement Method Using Product of Impedances

By taking the product of the input impedance under short-circuit conditions (Z_sc = Z0 tanh(?l)) and the input impedance under open-circuit conditions (Z_oc = Z0 coth(?l)), one can derive a relation that equals the square of the characteristic impedance. This method provides a practical way to measure the characteristic impedance of the line by experimentally determining both input impedances and computing Z0 using the relation Z0 = ?(Z_sc · Z_oc).

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